Theorem pm5.32da 622

Description: Distribution of implication over biconditional (deduction rule). (Contributed by NM, 9-Dec-2006.)

Hypothesis

Ref	Expression
pm5.32da.1	|- ((ph /\ ps) -> (ch <-> th))

Assertion

Ref	Expression
pm5.32da	|- (ph -> ((ps /\ ch) <-> (ps /\ th)))

Proof of Theorem pm5.32da

Step	Hyp	Ref	Expression
1		pm5.32da.1	. . 3 |- ((ph /\ ps) -> (ch <-> th))
2	1	ex 423	. 2 |- (ph -> (ps -> (ch <-> th)))
3	2	pm5.32d 620	1 |- (ph -> ((ps /\ ch) <-> (ps /\ th)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  rexbida  2629  reubida  2793  rmobida  2798  fcnvres  5243  funbrfv2b  5362  dffn5  5363  fnrnfv  5364  fniniseg  5371  eqfnfv2  5393  funiunfv  5467  dff13  5471  mpteq12f  5655  mpt2eq3dva  5669  ltlenlec  6207